On the asymptotic expansion for the relative Reshetikhin-Turaev invariants of fundamental shadow link pairs

Abstract

We study the asymptotic expansion conjecture of the relative Reshetikhin-Turaev invariants proposed in \cite{WY4} for all pairs $(M,L)$ satisfying the property that $M\setminus L$ is homeomorphic to some fundamental shadow link complement. The hyperbolic cone structure of such $(M,L)$ can be described by using the logarithmic holonomies of the meridians of some fundamental shadow link. We show that when the logarithmic holonomies are sufficiently small and all cone angles are less than $π$, the asymptotic expansion conjecture of $(M,L)$ is true. Especially, we verify the asymptotic expansion conjecture of the relative Reshetikhin-Turaev invariants for all pairs $(M,L)$ satisfying the property that $M\setminus L$ is homeomorphic to some fundamental shadow link complement, with cone angles sufficiently small. Furthermore, we show that if $M$ is obtained by doing rational surgery on a fundamental shadow link complement with sufficiently large surgery coefficients, then the cone angles can be pushed to any value less than $π$.